(linear algebra) A scalar, λ{\displaystyle \lambda \!}, such that there exists a vectorx{\displaystyle x} (the corresponding eigenvector) for which the image of x{\displaystyle x} under a given linear operatorA{\displaystyle {\rm {A\!}}} is equal to the image of x{\displaystyle x} under multiplication by λ{\displaystyle \lambda }; i.e. Ax=λx{\displaystyle {\rm {A}}x=\lambda x\!}

The eigenvaluesλ{\displaystyle \lambda \!} of a square transformation matrix M{\displaystyle {\rm {M\!}}} may be found by solving det(M−λI)=0{\displaystyle \det({\rm {M}}-\lambda {\rm {I}})=0\!} .

When unqualified, as in the above example, eigenvalue conventionally refers to a right eigenvalue, characterised by Mx=λx{\displaystyle {\rm {M}}x=\lambda x\!} for some right eigenvectorx{\displaystyle x\!}. Left eigenvalues, charactarised by yM=yλ{\displaystyle y{\rm {M}}=y\lambda \!} also exist with associated left eigenvectorsy{\displaystyle y\!}. For commutative operators, the left eigenvalues and right eigenvalues will be the same, and are referred to as eigenvalues with no qualifier.